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(Euler's approach was out of date by the middle 1840's owing
to the work of Gauss, Abel, and Cauchy), Biemann later became
the acute analyst that he did. But from Euler he may have
picked up something which also has its place in creative mathe-
matical work, an appreciation of symmetrical formulae and
manipulative ingenuity. Although Riemann depended chiefly
on what may be called deep philosophical ideas - those which
get at the heart of a theory - for his greater inspirations, his
work nevertheless is not wholly lacking in the 'mere ingenuity*
of which Euler was the peerless master and which it is now
quite the fashion to despise. The pursuit of pretty formulae and
neat theorems can no doubt quickly degenerate into a silly vice,
but so also can the quest for austere generalities which are so
very general indeed that they are incapable of application to
any particular. Riemann's instinctive mathematical tact pre-
served him from the bad taste of either extreme.
In 1S46, at the age of nineteen, Riemann matriculated as a
student of philology and theology at the University of Gottin-
gen. His desire to please his father and possibly help financially
by securing a paying position as quickly as possible dictated the
choice of theology. But he could not keep away from the mathe-
matical lectures of Stern on the theory of equations and on
definite integrals, those of Gauss on the method of least squares,
and Goldschmidt's on terrestrial magnetism. Confessing all to
his indulgent father, Riemann prayed for permission to alter
his course. His father's ungrudging consent that Bernhard
follow mathematics as a career made the young man supremely
happy - also profoundly grateful.
After a year at Gottingen, where the instruction was decid-
edly antiquated, Riemann migrated to Berlin to receive from
Jacobi, Diriehlet, Steiner, and Eisenstein his initiation into new
and vital mathematics. From all of these masters he learned
much - advanced mechanics and higher algebra from Jacobi,
the theory of numbers and analysis from Dirichlet, modem geo-
metry from Steiner, while from Eisenstein, three years older
than himself, he learned not only elliptic functions btrt self-
confidence, for he and the young master had a radical and roost
energizing difference of opinion as to how the theory should be